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International Journal of Scientific Engineering and Research (IJSER) www.ijser.in
ISSN (Online): 2347-3878, Impact Factor (2014): 3.05
Volume 3 Issue 10, October 2015 Licensed Under Creative Commons Attribution CC BY
Analysis of Fracture Parameters in Concrete by
Using ABAQUS Software
Krishna Kumar M1, Ganesh Naidu Gopu
2
1Pydah College of Engineering & Technology, vizag-530004, Andrapradesh, India
2Assistant Professor, Pace Institute of Technology and Sciences, NH-5, Near valluramma temple, Ongole-523001, Andrapradesh, India
Abstract: It is commonly accepted that there is a size effect on the nominal resistances of quasi-brittle materials such as cementitious
materials. This effect must be taken into account in the design of the ultimate behavior of concrete structures in order to avoid damage and
crack openings. These parameters are frequently used to study the behavior of concrete and to characterize the durability of structures.
Different theories exist in the literature to describe the size effect. Among them, we find the deterministic theory of Bazant where fracture
energy is considered independent of the size and it is assumed that at peak load, the crack length is proportional to the size of the specimen.
In this work, attention is paid to investigate numerically, the relationship between crack openings and length, and the size of the specimens.
Various fracture parameters have also been studied by validating with the existing work for a concrete of grade M35 grade of concrete
based on the RILEM standards. The present study shows the determination of fracture parameters of beams size-ranging from 100-400 mm
using ABAQUS. Then analyzed the size-effect behavior of various fracture parameters obtained from different sizes of the beam.
Keywords: Stress intensity factor, Strain energy release rate, Crack tip opening displacement, Fracture energy
1. Introduction
1.1. Fracture in Concrete
Fracture is a problem that society has faced for as long as there
have been man-made structures. The problem may actually be
worse today than in previous centuries, because more can go
wrong in our complex technological society. Fracture
mechanics is a branch of solid mechanics which deals with the
behavior of the material and conditions in the vicinity of a
crack and at the crack tip. The concept of linear elastic fracture
mechanics has been well developed for more than past 40
years and successfully applied to metallic structures. Concrete
is a heterogeneous anisotropic non-linear inelastic composite
material, which is full of flaws that may initiate crack growth
when the concrete is subjected to stress. Failure of concrete
typically involves growth of large cracking zones and the
formation of large cracks before the maximum load is reached.
This fact several properties of concrete, point toward the use
of fracture mechanics. Furthermore, the tensile strength of
concrete is neglected in most serviceability and limit state
calculations. Neglecting the tensile strength of concrete makes
it difficult to interpret the effect of cracking in concrete.
Different non-linear fracture models are established for
calculating the different fracture parameters. Those models are
Frictious crack model, Crack band model, Size effect model,
effective crack model, Two Parameter Fracture Model, Double
G Fracture model, Double K fracture Model, Smeared Crack
model, Discrete Crack model etc. The main fracture
parameters can calculate from the models are Stress Intensity
Factor, Crack Tip Opening Displacement and Strain Energy
Release Rate etc.
1.2. Size Effect in Fracture
The influence of size effect on concrete structures has been a
challenging issue during the recent past considering the fact
that, the raw materials as well as type of concrete adopted. The
size effect in quasi-brittle materials such as concrete is a well-
known phenomenon and there are a number of experimental
and theoretical studies. The size-effect is the decrease in
nominal strength of geometrically similar structures subjected
to symmetrical loads when the characteristic size of the
structure is increased. There are two extremes of size-effect
law as (i) strength criteria and (ii) LEFM size-effect. The
former yields no size-effect whereas the latter shows the
strongest size-effect i.e. nominal strength is inversely
proportional to the square root of the structural dimension.
LEICESTER seems to have been first to investigate the effect
of size on the structures made of metals, timber and concrete.
In order to illustrate size dependence in a simple and
dimensionless way, Hillerborg introduced the concept of
characteristic length. As a unique material property, the
characteristic length expresses fracture of concrete and
concrete like materials, where the characteristic length (lch )
proportional toft2. This means that brittleness increases with
an increase in the strength of concrete, but it decreases with
high fracture energy, according to fractious crack model
(F.C.M).
1.3. Introduction of FEM and ABAQUS
The finite element analysis is a numerical technique. In this
method all the complexities of the problems, like varying
shape, boundary conditions and loads are maintained as they
are but the solutions obtained are approximate. It started as an
extension of matrix method of structural analysis. Today this
method is used not only for the analysis in solid mechanics,
but even in the analysis of fluid flow, heat transfer, electric
and magnetic fields and many others. Civil engineers use this
method extensively for the analysis of beams, space frames,
plates, shells, folded plates, foundations, rock mechanics
problems and seepage analysis of fluid through porous media.
Both static and dynamic problems can be handled by finite
element analysis. the FEM is a highly suited method for
approximating the solution to the differential equation
governing the addressed problem. . Some popular Finite
Element packages are STAAD-PRO, GT-STRUDL,
NASTRAN, ABAQUS and ANSYS. Using these packages
one can analyze several complex structures.
Paper ID: IJSER15534 108 of 113
International Journal of Scientific Engineering and Research (IJSER) www.ijser.in
ISSN (Online): 2347-3878, Impact Factor (2014): 3.05
Volume 3 Issue 10, October 2015 Licensed Under Creative Commons Attribution CC BY
A popular enriched method is the so-called extended finite
element method, abbreviated XFEM. The XFEM was
implemented by Dassault Systemes Simulia Corp. [2010] in
their latest version of ABAQUS 6.10, which puts the engineer
in a position of being able to qualitatively estimate crack
patterns, spacing and widths for arbitrary geometries. A
complete ABAQUS analysis usually consists of three distinct
stages: preprocessing, simulation, and post processing.
Discrete crack modeling concept in ABAQUS aims at the
initiation and propagation of dominating cracks, whereas the
smeared crack model is based on the observation, that the
heterogeneity of concrete leads to the formation of many,
small cracks which, only in a later stage, nucleates to form one
larger, dominant crack.
Fanella and Krajcinovic [4] studied the effect of size on the
rupture strength of plain concrete by focusing on the
phenomena in the meso-scale of the material. Bazant and
Kazemi [2] reformulated Bazant's size effect law applied to
rock and concrete for the nominal stress at failure in a manner
in which the parameters are the fracture energy and the
fracture process zone length. From experimental as well as
TPCM model they observed that, for three-point bend beams,
TPFM predicts that the nominal strength decreases with
increasing beam size, but approaches to a minimum constant
value when sizes of the beam become very large. Tang, Shah,
et al., [3] used Two Parameter Concrete Model (TPCM) to
study the size effect of concrete in tension. Kotsovost and
Pavlov [9] investigated the causes of size effects in structural-
concrete experimentally and computationally. Their results
demonstrated that the finite element package used can also
provide a close fit to experimental values.
Rios, Jorge and Riera [8] conducted experiments as well as
introduced the concept of Discrete Element Modeling (DEM)
approach for studying the size effect in the analysis of
reinforced concrete structures. In their work they have taken 4
different specimens by varying the dimensions of the beam
and observed crack patterns under applied loading conditions
for each of the four specimens. Alam, Kotronis, et al., [1]
conducted experimental and numerical investigations on the
influence of size effect on crack opening, crack length and
crack propagation. Results obtained have not shown the
accuracy in using the proposed model effectively for Size
Effect phenomenon compared to that of experimental results.
Muralidhara Rao, Gunneswara Rao [6] conducted tests on 45
beams of geometrically similar notched plain concrete (M25)
specimens of different sizes. Fracture energy calculated using
Work-of-fracture method was increasing with the increase in
size of specimen and decreasing with the increasing notch
depth ratios.
The size-effect relationships between FCM, SEM and TPFM
were developed by Planas and Elices [5] (1990) that predicted
almost the same fracture loads for practical size range (100
400 mm) of precracked concrete beam for TPBT geometry.
Ouy ang et al. [10] (1996) established an equivalency between
TPFM and SEM based on infinitely large size specimens. It
was found that the relationship between CTODcs and cf
theoretically depends on both specimen geometry and initial
crack length and both the fracture models can reasonably
predict fracture behavior of quasi-brittle materials. Hanson and
Ingraffea [7] (2003) developed the size-effect, two-parameter,
and fictitious crack models numerically to predict crack
growth in materials for three-point bend test. The investigation
showed that if the three models must predict the same
response for infinitely large structures, they do not always
predict the same response on the laboratory size specimens.
Roesler et al. [11] (2007) plotted the size-effect behavior of
experimental results, numerical simulation using cohesive
crack model, size-effect model and two parameter fracture
model for three-point bend test specimens. From the analysis
of results it is found that the size-effect behavior calculated
from SEM and TPFM resembles closely.
Shailendra Kumar and S.V.Barai [12] presented the numerical
study on two parameter fracture model, effective crack model,
size effect model, double-k and double-g models. They found
that The critical stress intensity factors obtained using SEM,
ECM, DKFM and DGFM appear to be close to each other
with an error range of ±20%, TPFM predicted the most
conservative critical stress intensity factor. The crack-tip
opening displacement at unstable fracture load predicted using
TPFM was more conservative than that predicted using
DKFM or DGFM by about in the range of 27-47%.
2. Finite Element Modeling of Concrete Beams
2.1. Material Properties and Geometry
Standard specimens of three-point bending test as shown in
Fig. 2 are developed in the present study. Effect of self-weight
of the beam is also considered in the numerical model. The
influence coefficients of the COD equation are determined
using linear elastic finite element method. The same grade of
the concrete taken by Shailendra Kumar and S.V. Barai [33] in
2012 is taken in the present investigation as M35. The direct
local tensile strength of the concrete ft is taken as 3.21 Mpa.
The modulus of elasticity of concrete (E) is taken as 30 Gpa
and fracture energy (GF) is taken as 0.103 N/mm. The value of
Poisson ratio (ⱴ) is assumed to be 0.18. For TPBT specimen of
notched concrete beam with B = 100 mm, size range 100 ≤ D
≤ 400 mm and aspect ratio(S/D) ranging between 3-6, the
finite element analysis is carried out for determining the
fracture peak load using fictitious crack model at initial crack
length/depth (ao/D) ratios ranging between 0.2-0.5.
Figure 3.1: Three point bending test (TPBT) specimen
geometry
In Fractious crack model the stress ranges between ft and 1
3ft .
An equation derived for calculating the stress 𝜎 from the
stress-strain softening curve.
The equation
σ = ft 1 − 3w
w f
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The terminal point of the softening curve is denoted aswf , the
crack width is noted by w. From these stress value we can
found out the ultimate load (P) from the equation
P =1.072B(1−a0)
2σ
S
Supports adopted were simply supported and the load is
considered to be concentrated load acting at the centre of the
beam. Notch depth was considered for analysis. The notch
depths are taken as seam in vertical direction in the ABAQUS
at the bottom of the beam exactly under the point load taken.
Different notch depths were taken for the estimation of various
fracture parameters. The crack face is taken as the terminal
line of the notch. The extension direction is taken as q vector
which is taken before the notch face. Here the singularity is
taken as 0.25.
2.2. Mesh Size and Meshing Element
Generate the finite element mesh by choosing the meshing
technique that ABAQUS/CAE can use to create the mesh, the
element shape, and the element type. In the assigning section
we can assign a particular ABAQUS element type to the
model. Basic meshing is a two-stage operation: first seed the
edges of the part instance, and then mesh the part instance.
The number of seeds are based on the desired element size or
on the number of elements that along an edge.
Seeding was done with three different element sizes. At the
mid span of the beam very fine mesh has to be adopted, and
mesh size gets reduced moving towards the supports. For
L/D=4 from support to a distance of D has been taken with
course mesh. Then for 0.75D length has been taken medium
mesh. At the mid span of length 0.25D has been taken very
fine mesh. Structured hexagonal element has to be taken for
meshing. See the figure below
Figure 3.2: Seeding and meshing of the beam 300×1200
The quadratic reduced-integration elements available in
ABAQUS/Standard also have hourglass modes. However, the
modes are almost impossible to propagate in a normal mesh
and are rarely a problem if the mesh is sufficiently fine. The 1
× 6 mesh of C3D20R elements fails to converge because of
hour glassing unless two elements are used through the width,
but the more refined meshes do not fail even when only one
element is used through the width. Quadratic reduced-
integration elements are not susceptible to locking, even when
subjected to complicated states of stress. Therefore, these
elements are generally the best choice for most general
stress/displacement simulations, except in large-displacement
simulations involving very large strains and in some types of
contact analyses.
2.3. Output
The fracture parameters discussed are mainly are stress
intensity factor (K), strain energy release rate (Gf), crack tip
opening displacement (CTOD) and length of fracture process
zone. From the fracture analysis of the beam in ABAQUS we
can get the J- integral which is called strain energy release rate
in LEFM, stress intensity factor(K) for max strain energy
release rate in mode I. generally the outputs are obtained in the
.DAT, .FIL, .ODB and .PRT files. The stress intensity factor
and strain energy release rate values are obtained in the .DAT
file, and the stresses, strains, displacements and reaction forces
are obtained in the .FIL file.
Figure 3.3: Stresses formed in the deformed beam in 3D
Figure 3.4: Stresses formed in the deformed beam in 2D
3. Results and Discussions
3.1. Validation and Comparison with Reference [12]
Stress intensity factor (K) was calculated from the finite
element analysis using ABAQUS and CTOD value was
calculated from equations. The obtained results of Discrete
Crack Modeling (DCM) in table 2 are compared with results
of Sailendra Kumar and S.V.Barai [33].
3.1.1. Size effect behavior of stress intensity factor of
different models
Figure 4.1 shows the stress intensity factor obtained from
different models with respect to the size of the beam for
a/D=0.2. From the figures it clearly shows that the stress
intensity factor varies with depth of the specimen. The stress
intensity factor increases with increase in the depth of the
beam. It shows the size effect occurs in the fracture of
concrete. From the figures it can be observed that the stress
intensity factor KICe of effective crack model(ECM) shows the
higher values. Double K and double G models values KICun and
KICun are almost equal with small variation and shows almost
same size effect behaviour. The stress intensity factor KI of
calculated discrete crack model are nearly equal to the two
parameter fractre model (TPFM) results KICS .This means that
TPFM and DCM predict the most conservative results of
critical stress intensity factor at unstable failure. When the
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notch depth increases the K values obtained by calculated
DCM slightly comes near to the K values obtained by TPFM.
So the parameter K obtained from DCM was very similar to
the parameter K obtained from TPFM.
Figure 4.1: Size effect behavior of stress intensity factor from
different models for a/D=0.2
3.1.2. Size effect behavior of crack tip opening
displacement (CTOD)
The crack tip opening displacement (CTOD) of DCM obtained
analytically was compared with CTODCS of TPFM and
CTODCof Double K model obtained from the reference the
figures 4.2 and 4.3 drawn for analysis.
Figure 4.2: Size effect behavior of crack tip opening
displacement of different models for a/D=0.2
Figure 4.3: Relationship of the CTODCS and CTODD obtained
between using TPFM and DCM
From the fig 4.2 that the CTODD , CTODCS and CTODC maintain
a definite relationship with the specimen size for a given value
of ao/D ratio and they increase as the specimen size increases.
It is also observed from the fig, the CTODCS and CTODC
depend on the a/D ratio for a given specimen size. Here
CTODCS of TPFM having lesser values compared to remaining
values. The CTODD of DCM is nearer to the CTODC of Double
k model. As notch depth increases the CTODD values obtained
from DCM increase and slightly reaches to the CTODCSof
Double K model. TheCTODCS parameter values are more
scattered particularly for smaller size of specimens when
compared among the different ao/D ratios whereas the CTODC
values are more nearer and less scattered and appear to be in a
narrow band for size-range 100-400 mm considered in the
study.
A relationship between CTODCS and CTODD is presented in
Fig. 4.3 in which the ratio CTODCS /CTODD , is plotted with
respect to the depth if the beam D. It is seen from the figure
that the ratio CTODCS /CTODD maintains a definite relationship
with the specimen size and the ratio decreases as the specimen
size increases. Neglecting the effect of a/D ratio, the mean
values of CTODCS /CTODD for specimen sizes range 100 and
400 mm are determined and found to be 0.8425 and 0.57
respectively. It means that the predicted CTOD at critical load
using DCM is conservative than TPFM and Double K model.
3.2. Validation for Various Aspect Ratios of The Beam:
Using the results and with reference to [35] the work has been
extended to Plain concrete beams of different geometry with
different aspect ratios L/d from 4 to 6 for a/D=0.2-0.5
analyzed with discrete crack method in 3 dimensional finite
element analysis using ABAQUS 6.10. The following fracture
parameters have been calculated for different sizes of beams
with different aspect ratios for different notch depths.
1. Stress intensity factor (K).
2. Crack tip opening displacement (CTOD).
3. Strain energy release rate (Gf).
3.2.1. Size effect behavior of stress intensity factor for
various SENB beams of different sizes
The stress intensity factor values of various sizes of beams
were calculated using three-dimensional finite element
analysis of fractured beam.
3.2.1.1. Variation of stress intensity factor with various
beam sizes:
Some graphs were drawn for analyze the size effect behavior
of fracture parameter stress intensity factor (K). Here the stress
intensity factor (K) varies with respect to the size of the beam.
The figure 4.4 shows the stress intensity factor variation with
the size of the beams.
Figure 4.4: Size of the beam vs stress intensity factor of
SENB beams
The fig 4.4 clearly shows that stress intensity factor increases
with the size of the beam. When observed the figures the line
becomes flatter when the size of the beam increases. It means
the stress intensity factor values become closer when the size
of the beam increases. So it shows the size effect of stress
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intensity facture disappear when the size of the beam
increases.
3.2.1.2. Variation of stress intensity factor with various
beam lengths
Figure 4.5 drawn that the stress intensity factor variation with
the length of the beams.
Figure 4.5: Length vs stress intensity factor of SENB beams
of depth 100mm
In fig 4.5 the lines were straight with constantly increasing
values. It shows that the stress intensity factor varies lightly
but constantly with the length of the beam. So it shows that
that the size effect of stress intensity factor in length is less but
constantly moving with the length of the beam.
3.2.1.3. Variation of stress intensity factor with various
depths of the beam
Fig 4.6 shows the stress intensity factor variation with the
depth of the beams.
Figure 4.6: Depth vs stress intensity factor in SENB beam of
length 1200 mm
The fig 4.6 showed clearly the stress intensity factor increases
with the depth of the beam. the lines become flatter when the
depth increases of the same length 1200 mm of the beam. So it
clearly showed the size effect of stress intensity factor
parameter constantly moving when depth of the beam
increases for lesser beam lengths. For higher length beams the
size effect of stress intensity factor disappears when the depth
increases.
3.2.1.4. Variation of stress intensity factor with various
notch depths of the beam
Figure 4.7: Notch depth ratio vs stress intensity factor in
SENB beams
Figure 4.7 showed the stress intensity factor variation with the
notch-depth ratio. From graphs, it can observe that that stress
intensity factor increases with the notch depth also. There was
very small variation in the stress intensity factor in large notch
depth ratio of the beam. So size effect of stress intensity factor
is less when the notch depth of the beam increases.
3.2.2. Size effect behavior of various crack tip opening
displacement (CTOD) for various SENB beams of
different sizes:
Crack tip opening displacement calculated from stress
intensity factor and strain energy release rate using numerical
equations.
The crack tip opening displacement increases with the size of
the beam. It means the Crack tip opening displacement
becomes equal when the size of the beam increases. So it
shows the size effect of Crack tip opening displacement
disappears when the size of the beam increases.
The crack tip opening displacement increases with the length
of the beam. It clearly means that the crack tip opening
displacement varies lightly but constantly with the length of
the beam. So it shows that that the size effect of crack tip
opening displacement in length is less but constantly moving
with the length of the beam.
The crack tip opening displacement increases constantly with
depth of the beam. The CTOD parameter variation is less
when the depth increases of having higher lengths compared
to lower lengths. So the size effect behavior of CTOD is less
when the depth increases for the higher lengths of the beam
compared o the lower lengths of the beam.
The CTOD increases with the notch depth also. A very small
variation in the CTOD occurs when notch depth of the beam
increases. So size effect of crack tip opening displacement is
less when the notch depth of the beam increases.
3.2.3. Size effect behavior of strain energy release rate (𝐆𝐟)
of various SENB beams of different sizes:
Strain energy release rate is called as j integral in non linear
fracture mechanics. This parameter calculated directly from
the finite element analysis using ABAQUS.
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The strain energy release rate increases with the size of the
beam. The strain energy release rate becomes equal when the
size of the beam increases. So it shows the size effect of strain
energy release rate decreases when the size of the beam
increases.
The strain energy release rate increase with the length of the
beam. the strain energy release rate varies lightly but
constantly with the length of the beam. So it shows that that
the size effect of strain energy release rate in length is less but
constantly moving with the length of the beam.
The strain energy release rate increases with the depth of the
beam of constant length having different notch depths. the
variation of strain energy release rate is less when the depth
increases of having higher lengths compared to lower lengths.
So the size effect behavior of strain energy release rate is less
when the depth increases for the higher lengths of the beam
compared o the lower lengths of the beam.
The strain energy release rate increases with the notch depth
also. It shows the lines are very flatter. It means there is very
small variation in the strain energy release rate with notch
depth ratio of the beam. So size effect of strain energy release
rate is less when the notch depth of the beam increases.
4. Conclusions
In the present study the size-effect analysis of various fracture
parameters obtained from different sizes of the beam were
calculated. The fracture parameters were determined from
three-dimensional finite element analysis using ABAQUS
6.10 in discrete crack modeling of size-ranging from 100-400
mm for which the input data were obtained from cohesive
crack model. A comparative size-effect study was carried out
using the possible fracture parameters from DCM, TPFM,
SEM, ECM, DKFM and DGFM. In general, it was observed
that all the fracture parameters were dependent on geometrical
factor and specimen size. From present numerical study the
following remarks can be highlighted.
i. All fracture parameters obtained from numerical
analysis of different sizes of beams are exhibiting the
size effect phenomenon.
ii. The critical stress intensity factor obtained from
numerical analysis gave proximate results when
compared to the stress intensity factor obtained from
TPFM.
iii. The crack tip opening displacement obtained from
numerical analysis gave proximate results when
compared to the Double K model and TPFM. But when
continuously increasing sizes of the beam the crack tip
opening displacement values of numerical analysis
equals to the crack tip opening displacement of Double
K model.
iv. When the size of the beam increases, the size effect
behavior of fracture parameters are increases till some
point and then remains constant which shows that the
size effect behavior gets disappears.
v. When the length of the beam increases, the size effect
behavior of all fracture parameters is small compared to
other dimensions, but varies constantly.
vi. When the depth of the beam increases, the size effect
behavior of all fracture parameters is large compared to
other dimensions, but this behavior disappears while
continuously increasing the depth.
vii. When the notch depth increases, then the size effect
behavior of fracture parameters increases.
Acknowledgement
All authors are grateful to thank to the management of
Sathyabama University, Chennai and BITS Hyderabad for the
facility provided to complete the present study.
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